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1 ISSN Working Paper Series Selection of Optimal Lag Length in Cointegrated VAR Models with Weak Form of Common Cyclical Features Carlos Enrique Carrasco Gutiérrez, Reinaldo Castro Souza and Osmani Teixeira de Carvalho Guillén June, 2007

2 ISSN CGC / Working Paper Series Brasília n. 139 Jun 2007 P. 1-35

3 Working Paper Series Edited by Research Department (Depep) Editor: Benjamin Miranda Tabak Editorial Assistent: Jane Sofia Moita Head of Research Department: Carlos Hamilton Vasconcelos Araújo The Banco Central do Brasil Working Papers are all evaluated in double blind referee process. Reproduction is permitted only if source is stated as follows: Working Paper n Authorized by Mário Mesquita, Deputy Governor for Economic Policy. General Control of Publications Banco Central do Brasil Secre/Surel/Dimep SBS Quadra 3 Bloco B Edifício-Sede M1 Caixa Postal Brasília DF Brazil Phones: (5561) and Fax: (5561) The views expressed in this work are those of the authors and do not necessarily reflect those of the Banco Central or its members. Although these Working Papers often represent preliminary work, citation of source is required when used or reproduced. As opiniões expressas neste trabalho são exclusivamente do(s) autor(es) e não refletem, necessariamente, a visão do Banco Central do Brasil. Ainda que este artigo represente trabalho preliminar, citação da fonte é requerida mesmo quando reproduzido parcialmente. Consumer Complaints and Public Enquiries Center Address: Secre/Surel/Diate Edifício-Sede 2º subsolo SBS Quadra 3 Zona Central Brasília DF Brazil Fax: (5561) Internet:

4 Selection of Optimal Lag Length in Cointegrated VAR Models with Weak Form of Common Cyclical Features Carlos Enrique Carrasco Gutiérrez Reinaldo Castro Souza Osmani Teixeira de Carvalho Guillén The Working Paper should not be reported as representing the views of the Banco Central do Brasil. The views expressed in the papers are those of the author(s) and not necessarily reflect those of the Banco Central do Brasil. Abstract An important aspect of empirical research based on the vector autoregressive (VAR) model is the choice of the lag order, since all inference in the VAR model depends on the correct model specification. Literature has shown important studies of how to select the lag order of a nonstationary VAR model subject to cointegration restrictions. In this work, we consider an additional weak form (WF)restrictionofcommoncyclicalfeaturesinthemodelinorder to analyze the appropriate way to select the correct lag order. Two methodologies have been used: the traditional information criteria (AIC, HQ and SC) and an alternative criterion (IC(p, s)) which select simultaneously the lag order p and the rank structure s due to the WF restriction. A Monte- Carlosimulationisusedintheanalysis. The results indicate that the cost of ignoring additional WF restrictions in vector autoregressive modelling can be high specially when SC criterion is used. Keywords: Cointegration; Common Cyclical Features; Reduced Rank Model; Estimation; Information Criteria. JEL Codes: C32, C53. Acknowledgments: We are grateful to comments and suggestions given by João Victor Issler, Wagner Gaglianone, Ricardo Cavalcanti, Luiz Renato Lima and participants of the Brazilian Econometric Meeting Special thanks are due to Alain Hecq for solving doubts and comments. The authors are responsible for any remaining errors in this paper. Carlos Enrique C. Gutiérrez acknowledges the support of CAPES-Brazil. Corresponding author, Graduate School of Economics, FGV-EPGE-RJ, Brazil. DEE-PUC-RJ, Brazil Banco Central do Brasil and Ibmec-RJ, Brazil 3

5 1 Introduction In the modelling of economic and financial time series, the vectorial autoregressive (VAR) model became a standard linear model used in empirical works. An important aspect of empirical research in the specification of the VAR models is the determination of the lag order of the autoregressive lag polynomial, since all inference in the VAR model depends on the correct model specification. In several contributions, the effect of lag length selection has been demonstrated: Lütkepohl (1993) indicates that selecting a higher order lag length than the true lag length causes an increase in the mean square forecast errors of the VAR and that underfitting the lag length often generates autocorrelated errors. Braun and Mittnik (1993) show that impulse response functions and variance decompositions are inconsistently derived from the estimated VAR when the lag length differs from the true lag length. When cointegration restrictions are considered in the model, the effect of lag length selection on the cointegration tests has been demonstrated. For example, Johansen (1991) and Gonzalo (1994) point out that VAR order selection may affect proper inference on cointegrating vectors and rank. Recently empirical works have considered another kind of restrictions on the VAR model (e.g., Engle and Issler, 1995; Caporale, 1997; Mamingi and Sunday, 2003). Engle and Kozicki (1993) showed that VAR models can have another type of restrictions, called common cyclical features, which are restrictions on the short-run dynamics. These restrictions are defined in the same way as cointegration restrictions, while cointegration refers to relations among variables in the long-run, the common cyclical restrictions refer to relations in the short-run. Vahid and Engle (1993) proposed the Serial Correlation Common Feature (SCCF) asameasureof common cyclical feature. SCCF restrictions might be imposed in a covariance stationary VAR model or in a cointegrated VAR model. When short-run restrictions are imposed in cointegrated VAR models it is possible to define a weak version of SCCF restrictions. Hecq, Palm and Urbain (2006) definedaweakversion ofsccf restrictions which they denominated it as weak-form (WF) common cyclical restric- 4

6 tions. A fundamental difference between SCCF and WF restrictions is in the form which each one imposes restrictions on the Vector Error Correction Model (VECM) representation 1. When SCCF are imposed, all matrices of a VECM have rank less than the number of variables analyzed. On the other hand with WF restrictions all matrices, except the long-run matrix, have rank less than a number of variables in analysis. Hence, WF restrictions impose less restriction on VECM parameters. Some advantages emerge when WF restrictions are considered. First, due to the fact that WF restrictions does not impose restrictions on the cointegration space; the rank of common cyclical features is not limited by the choice of cointegrating rank. Another advantage is that WF restrictions is invariant over reparametrization in VECM representation. The literature has shown how to select an adequate lag order of a covariance stationary VAR model and an adequate lag order of a VAR model subject to cointegration restrictions. Among the classical procedures, there are the information criteria such as Akaike (AIC), Schwarz (SC) and Hannan-Quinn (HQ) (Lütkepohl, 1993). Kilian (2001) study the performance of traditional AIC, SC and HQ criterion of a covariance stationary VAR model. Vahid and Issler (2002) analyzed the standard information criterion in a covariance stationary VAR model subject to SCCF restriction and more recently Guillén, Issler and Athanasopoulos (2005) studied the standard information criterion in VAR models with cointegration and SCCF restrictions. However, when cointegrated VAR models contain additional weak form of common cyclical feature, there are no reported work on how to appropriately determine the VAR model order. The objective of this paper is to investigate the performance of information criterion in selecting the lag order of a VAR model when the data are generated from a true VAR with cointegration and WF restrictions that is referred as the correct model. It will be carried out following two procedures: a) the use of standard criteria as proposed by Vahid and Engle (1993), referred here as IC (p), andb) the use of an 1 When a VAR model has cointegration restriction it can be represented as a VECM. This representation is also known as Granger Representation Theorem (Engle and Granger, 1987). 5

7 alternative procedure of model selection criterion (see, Vahid and Issler, 2002; Hecq et al., 2006) consisting in selecting simultaneously the lag order p and the rank s do to the weak form of common cyclical feature, which is referred to as IC(p, s) 2.The most relevant results can be summarized as follows. The information criterion that selects simultaneously the pair (p, s) has better performance than the model chosen by conventional criteria. The cost of ignoring additional WF restrictions in vector autoregressive modelling can be high specially when SC criterion is used. The remaining of this work is organized as follows. Section 2 shows the econometric model. In section 3 the information criteria are mentioned. Monte Carlo simulation is shown in section 4 and the results in section 5. Finally, the conclusions are shown in section 6. 2 The Econometric Model We show the VAR model with short-run and long-run restrictions. First, we consider a Gaussian vector autoregression of finite order p, so-called VAR(p), such that: y t = px A i y t i + ε t (1) i=1 where, y t is a vector of n first order integrated series, I(1), A i, i =1,...,p are matrices of dimension n n, ε t Normal(0, Ω) and {Ω, if t = τ and 0 n n, if t 6= τ, whereω is non singular}. The model (1) could be written equivalently as; Π (L) y t = ε t where L represents the lag operator and Π (L) =I n P p i=1 A il i that when L =1, Π (1) = I n P p i=1 A i. If cointegration is considered in (1) the (n n) matrix Π ( ) satisfies two conditions: a) Rank (Π (1)) = r, 0 <r<n, such that Π (1) can be expressed as Π (1) = αβ 0,whereα and β are (n r) matrices with full column rank, r. b) The characteristic equation Π (L) =0has n r roots equal to 1 and all other are outside the unit circle. These assumptions imply that y t is cointegrated of order (1, 1). The elements of α are the adjustment coefficients and the columns of β span the space of cointegration vectors. We can represent a VAR 2 This is quite recent in the literature (see, Hecq et al., 2006). 6

9 That is, β n (n s) is a full column rank orthogonal to the complement of β with rank( β, β )=n. Rewriting model (3) we have: y t = αβý t 1 + β (Ψ 1, Ψ 2,..., Ψ p 1 ) X t 1 + ε t (4) = αβý t 1 + β ΨX t 1 + ε t (5) Estimation of (5) is carried out via the switching algorithms (see, Hecq, 2006) that use the procedure in estimating reduced-rank regression models suggested by Anderson (1951). There is a formal connection between a reduced-rank regression and the canonical analysis as noted by Izenman (1975), Box and Tiao (1977), Tso (1980) and Veleu et al. (1986). When the multivariate regression has all of its matrix coefficients of full rank, it may be estimated by usual Least Square or Maximum- Likelihood procedures. But when the matrix coefficients are of reduced-rank they have to be estimated using the reduced-rank regression models of Anderson (1951). The use of canonical analysis may be regarded as a special case of reduced-rank regression. More specifically, the maximum-likelihood estimation of the parameters of the reduced-rank regression model may result in solving a problem of canonical analysis 3. Therefore, we can use the expression CanCorr{X t,z t W t } that denotes the partial canonical correlations between X t and Z t : both sets concentrate out the effect of W t that allows us to obtain canonical correlation, represented by the eigenvalues ˆλ 1 > ˆλ 2 > ˆλ 3... > ˆλ n. The Johansen test statistic is based on canonical correlation. In model (2) we can use the expression CanCorr{ y t,y t 1 W t } where W t =[ y t 1, y t 2,..., y t+p 1 ] that summarizes the reduced-rank regression procedure used in the Johansen approach. It means that one extracts the canonical correlations between y t and y t 1 : both sets concentrated out the effect of lags of W t. In order to test for the significance of the r largest eigenvalues, one can rely on Johansen s trace statistic (6): nx ξ r = T Ln (1 ˆλ 2 i ) i =1,..., n (6) i=r+1 3 This estimation is referred as Full Information Maximum Likelihood -FIML 8

10 where the eigenvalues 0 < ˆλ n <... < ˆλ 1 are the solution of : λm 11 m 1 10 m 00 m 01 = 0, wherem ij, i, j =0.1, are the second moment matrices: m 00 = 1 T P T t=1 ũ0tũ 0 0t, m 10 = 1 T P T t=1 ũ1tũ 0 0t, m 01 = 1 T P T t=1 ũ0tũ 0 1t, m 11 = 1 T P T t=1 ũ1tũ 0 1t of the residuals ũ 0t and ũ 1t obtained in the multivariate least squares regressions y t =( y t 1,... y t p+1 )+ u 0t and y t 1 =( y t 1,... y t p+1 )+u 1t respectively (see, Hecq et al., 2006; Johansen, 1995). The result of Johansen test is a superconsistent estimated β. Moreover, we could also use a canonical correlation approach to determine the rank of the common features space due to WF restrictions. It is a test for the existence of cofeatures in the form of linear combinations of the variables in the first differences, corrected for long-run effects which are white noise (i.e., β 0 ( y t αβý t 1 )= β 0 ε t where β 0 ε t is a white noise). Canonical analysis is adopted in the present work in estimating, testing and selecting lag-rank of VAR models as shown in next sections. 3 Model Selection Criteria In model selection we use two procedures to identify the VAR model order. The standard selection criteria, IC(p) and the modified informational criteria, IC(p, s), novelty in the literature, which consists on identifying p and s simultaneously. The model estimation following the standard selection criteria, IC(p), used by Vahid and Engle (1993) entails the following steps: 1. Estimate p using standard informational criteria: Akaike (AIC), Schwarz (SC) and Hanna-Quinn (HQ). We choose the lag length of the VAR in levels that minimize the information criteria. 2. Usingthelaglengthchoseninthepreviousstep,find the number of cointegration vector, r using Johansen cointegration test Conditional on the results of cointegration analysis, a final VECM is estimated and then the multi-step ahead forecast is calculated. 4 Cointegration rank and vectors are estimated using the FIML as shown in Johansen (1991). 9

11 The above procedure is followed when there is evidence of cointegration restrictions. We check the performance of IC(p) when WF restrictions contain the true model. Additionally we check the performance of alternative selection criteria IC(p, s). Vahid and Issler (2002) analyzed a covariance-stationary VAR model with SCCF restrictions. They showed that the use of IC(p, s) has better performance than IC(p) in VAR model lag order selection. In the present work we analyze cointegrated VAR model with WF restrictions in order to analyze the performance of IC(p) and IC(p, s) for model selection. The question investigated is: is the performance of IC(p, s) superior to that of IC(p)? Thisisanimportantquestionweaim to answer in this work. The procedure of selecting the lag order and the rank of the structure of shortrun is carried out by minimizing the following modified information criteria (see Hecq, 2006). TX AIC (p, s) = ln(1 λ 2 i (p)) + 2 T N (7) i=n s+1 HQ(p, s) = TX i=n s+1 ln(1 λ 2 i (p)) + 2ln(lnT ) T N (8) SC(p, s) = TX i=n s+1 ln(1 λ 2 i (p)) + ln T T N (9) N =[n (n (p 1)) + n r] [s (n (p 1) + (n s))] The number of parameters N is obtained by subtracting the total number of mean parameters in the VECM (i.e., n 2 (p 1)+nr), for given r and p, fromthenumber of restrictions the common dynamics imposes from s (n (p 1)) s (n s). The eigenvalues λ i are calculated for each p. To calculate the pair (p, s) we assume that no restriction of cointegration exists, that is, r = n (see Hecq, 2006). We fix p in model (3) and then find λ i i =1, 2...n using the program cancorr( y t,x t 1 y t 1 ). This procedure is followed for every p and in the end we choose the p and s that 10

12 minimizes the IC(p, s). After selecting the pair (p, s) we can test the cointegration relation using the procedure of Johansen. Finally we estimate the model using the switching algorithms as shown in the next chapter. Notice that in this simultaneous selection, testing the cointegration relation is the last procedure to follow, so we are inverting the hierarquical procedure followed by Vahid and Engle (1993) where the first step is the selection of the number of cointegration relations. It may be an advantage specially when r is over-estimated. Few works have been dedicated to analyze the order of the VAR models considering modified IC(p, s). Asmentioned, Vahid and Issler (2002) suggested the use of IC(p, s) to simultaneously choose the order p andanumberofreduced rankstructures on covariance stationary VAR model subject to SCCF restrictions. However, no work has analyzed the order of the VAR model with cointegration and WF restrictions using a modified criterion, which is exactly the contribution of this paper. To estimate the VAR model considering cointegration and WF restrictions we use the switching algorithms model as considered by Hecq (2006). Consider the VECM given by: y t = αβ 0 y t 1 + β ΨX t 1 + ε t (10) A full description of switching algorithms is presented below in four steps: Step1:Estimation of the cointegration vectors β. Using the optimal pair ( p, s) chosen by information criteria (7), (8) or (9), we estimate β (and so its rank, r = r) using Johansen cointegration test. Step2:Estimation of β and Ψ. Taking ˆβ estimated in step one, we proceed to estimate β and Ψ. Hence, we run a regression of y t and of X t 1 on ˆβ 0 y t 1. We labeled the residuals as u 0 and u 1, respectively. Therefore, we obtain a reduced rank regression: u 0 = β Ψu 1 + ε t (11) 11

15 companion matrix. Hence, we follow an alternative procedure. We propose an analytical solution to generate a covariance-stationary VAR, based on the choice of the eigenvalues, and then on the generation of the respective companion matrix. In the appendix we present a detailed discussion of the final choice of these free parameters, including analytical solutions. In our simulation, we constructed 100 data generating processes and for each of these we generate 1000 samples containing 1000 observations. In order to reduce the impact of initial values, we consider only the last 100 and 200 observations. All the experiments were conducted in the MatLab environment. 5 Results Values in Table I represent the percentage of time that the model selection criterion, IC(p), chooses that cell corresponding to the lag and number of cointegration vectors in realizations. The true lag-cointegrating vectors are identified by bold numbers and the selected lag-cointegration vectors chosen more times by the criterion are underlined. The results show that, in general, the AIC criterion choose more frequently the correct lag length for 100 and 200 observations. For example, for 100 observations, the AIC, HQ and SC criteria chose the true lag, p, 54.08%, 35.62% and 17.49% of the times respectively. Note that all three criteria chose more frequently the correct rank of cointegration (r =1). When 200 observations are considered, the correct lag length was chosen 74.72%, 57.75% and 35.28% of the time for AIC, HQ and SC respectively. Again all three criteria selected the true cointegrated rank r =1. Tables II contains the percentage of time that the simultaneous model selection criterion, IC(p, s), chooses that cell, corresponding to the lag-rank and number of cointegrating vectors in 100,000 realizations. The true lag-rank-cointegration vectors are identified by bold numbers and the best lag-rank combination chosen more times by each criterion are underlined. The results show that, in general, the AIC criterion chooses more frequently lag-rank for 100 and 200 observations. For instance, for 100 observations, the AIC, HQ and SC criteria 14

16 choose more frequently the true pair (p, s) =(3, 1), 56.34%, 40.85% and 25.20% of the times respectively. For 200 observations, AIC, HQ and SC criteria choose more frequently the true pair (p, s) =(3, 1), 77.07%, 62.58% and 45.03% of the times respectively. Note that all three criteria choose more frequently the correct rank of cointegration (r =1)in both samples. The most relevant results canbesummarizedasfollows: All criteria (AIC, HQ and SC) choose the correct parameters more oftenwhenusingic(p, s). The AIC criterion has better performance in selecting the true model more frequently for both the IC(p, s) and the IC(p) criteria. When the size of the sample decreases the true value p is less frequently selected by all the traditional criteria. Table I shows that ignoring WF restrictions the standard SC has the worst performance in choosing the true value of p. It is known that literature suggests the use of the traditional SC and HQ criteria in VAR model selection. The results of this work indicate that if additional WF restrictions are ignored, the standard SC and HQ criteria select few times the true value of p. That is, there is a cost of ignoring additional WF restrictions in the model specially when SC criterion is used. In general, the standard Schwarz or Hannan- Quinn selection criteria should not be used for this purpose in small samples due to the tendency of identifying an underparameterized model. In general, the use of these alternative criteria of selection, IC(p, s) has better performance than the usual criteria, IC(p), when the cointegrated VAR model has additional WF restriction. 6 Conclusions In this work, we considered an additional weak form restriction of common cyclical features in a cointegrated VAR model in order to analyze the appropriate way for 15

17 selecting the correct lag order. These additional WF restrictions are defined in the same way as cointegration restrictions, while cointegration refers to relations among variables in the long-run, the common cyclical restrictions refer to relations in the short-run. Two methodologies have been used for selecting lag length; the traditional information criterion, IC(p), and an alternative criterion (IC(p, s)) that selects simultaneously the lag order p and the rank structure s due to the WF restriction. The results indicate that information criterion that selects the lag length and the rank order simultaneously has better performance than the model chosen by conventional criteria. When the WF restrictions are ignored there is a non trivial cost in selecting the true model with standard information criteria. In general, the standard Schwarz or Hannan-Quinn criteria selection criteria should not be used for this purpose in small samples due to the tendency of identifying an underparameterized model. In applied work, when the VAR model contains WF and cointegration restrictions, we suggest the use of AIC(p, s) criteria for simultaneously choosing the lagrank, since it provides considerable gains in selecting the correct VAR model. Since no work in the literature has been dedicated to analyze a VAR model with WF common cyclical restrictions, the results of this work provide new insights and incentives to proceed with this kind of empirical work. References [1] Anderson TW Estimating linear restrictions on regression coefficients for multivariate normal distributions. Ann. Math. Statist. 22: [Correction (1980) Ann. Statist ] [2] Bewley R, Yang M On the size and power of system tests for cointegration. The Review of Economics and Statistics 80:

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